UDC 517.946.6:517.946.9

ON BOUNDARY VALUE PROBLEMS FOR THE EQUATION \(\Delta^m u = 0\)

I. G. LYUBODZINSKAYA

The article considers an interior boundary value problem of general form for the equation \(\Delta^m u = 0\) in \(n\)-dimensional space. With the aid of certain \(m\)-harmonic potentials the problem is reduced to a system of integro-differential equations equivalent to a certain system of Fredholm integral equations of the second kind, and the existence of a solution of the problem is proved under the assumption of uniqueness.

The general scheme of the argument is as follows: suppose the problem has been reduced to a system of Fredholm integral equations of the second kind. Consider the corresponding homogeneous system. Let \(\mu_{j0}(Q)\) be its solution. Treating these as densities of potentials, we obtain that the sum of the potentials in the domain \(D\), where the solution is sought, is identically equal to zero by virtue of the uniqueness of the solution of the problem. Next this sum of potentials is considered in the exterior domain \(D_e\), where it satisfies the equation \(\Delta^m u = 0\) and certain homogeneous boundary conditions on the boundary. If the resulting exterior problem has a unique solution, then the sum of the potentials proves to be equal to zero in the whole space. Hence it follows that \(\mu_{j0}(Q)=0\), and by the Fredholm alternative the nonhomogeneous system of integral equations is solvable, and with it the original boundary value problem.

I. Let a certain bounded domain \(D\) with boundary \(\Sigma \in A_{2m+2}\) be given in \(n\)-dimensional space. Consider the boundary value problem. It is required to determine a function \(u(P)\) satisfying the following conditions:

\[ 1)\quad \Delta^m u = 0 \text{ in } D, \tag{1} \]

\[ 2)\quad \sum_{k=1}^{m+j} a_{hk}(P_0) u^{(k)}(P_0) = f_h(P_0) \text{ on } \Sigma \]

\[ (h=1,2,\ldots,m;\ j \leq m), \tag{2} \]

where

\[ u^{(2i)}(P_0)=\frac{\partial}{\partial \nu}\Delta^{i-1}u(P_0);\qquad u^{(2i+1)}(P_0)=\Delta^i u(P_0) \]

(\(\frac{\partial}{\partial \nu}\) denotes differentiation along the inward normal),

\[ 3)\quad u(P)\in H_{m+j}\text{ in }D+\Sigma \]

(\(H_{m+j}\) is the class of functions having bounded derivatives with respect to all arguments up to order \(m+j\) inclusive, which satisfy Hölder conditions).

We assume that the system of boundary conditions is compatible, that the rank of the matrix \((a_{hk})\) is equal to \(m\), and that at least one of the coefficients \(a_{h,m+j}\ne 0\). Such boundary conditions will be called conditions of order \(m+j\).

In order to systematize the boundary value problems under consideration, we introduce the notion of the class of a boundary condition.

Definition. A boundary condition of the form (2) is called a boundary condition of order \(r\), if it contains \(\Delta^{r-1}u(P_0)\) or

\[ \frac{\partial \Delta^{r-1}u(P_0)}{\partial \nu} \]

on the hypersurface \(\Sigma\), and contains no derivatives of higher orders. It is obvious that 1) \(r \leq m\); 2) one and the same problem may contain no more than two boundary conditions of one order; 3) the \((2l-1)\)-st boundary condition may be of order not lower than \(l\) and not higher than \(2l-1\), while the \(2l\)-th boundary condition, respectively, may be of order not lower than \(l\) and not higher than \(2l\) \(\left(l=1,2,\ldots,\left[\dfrac{m-1}{2}\right]\right)\).

Let the order of the \(i\)-th boundary condition be equal to \(r_i\); then the boundary value problem posed above will be denoted by the symbol \(m_{r_1,r_2,\ldots,r_m}\) (\(r_1\) is always equal to 1).

II. Let us construct a system of potentials for the equation \(\Delta^m u=0\). We divide all potentials into \(m\) classes. To the \(j\)-th class \((j=1,2,\ldots,m)\) we assign those which are solutions of the equation \(\Delta^j u=0\), but do not satisfy the equation \(\Delta^{j-1}u=0\). In all, the following \(2j\) potentials belong to the \(j\)-th class:

\[ u_{jh}(P)=\frac{1}{\pi}\int_{\Sigma}\mu_{jh}(Q)\frac{\cos^{2j-h}\Theta}{R^{n-h}}\,d\sigma_Q \quad (h=1,2,\ldots,j) \tag{3} \]

In addition to them, to the \(j\)-th class we assign the following \(j\) further potentials:

\[ u_{jh}(P)=\frac{1}{\pi}\int_{\Sigma}\mu_{jh}(Q)\varphi_{jh}(P,Q)\,d\sigma_Q \quad (h=j+1,\ldots,2j), \tag{4} \]

where

\[ \varphi_{jh}(P,Q)= \sum_{t=0}^{\,j-\left[\frac{h+1}{2}\right]} c_{jht}\, \frac{\partial^{2j-h-2t}}{\partial \nu^{2j-h-2t}} R^{2j-3-2t} + \sum_{t=0}^{\left[\frac{h-1}{2}\right]} d_{jht}R^{2t}. \tag{5} \]

The constants \(c_{jht}\) and \(d_{jht}\) are chosen in such a way that, for \(h=2k+1\),

\[ \Delta^k\varphi_{j,2k+1}(P,Q)= \frac{\cos^{2(j-k)-1}\Theta}{R^{n-1}}, \tag{6} \]

whereas if \(h=2k+2\), then

\[ \Delta^k\varphi_{j,2k+2}(P,Q)= \frac{\cos^{2(j-k)-2}\Theta}{R^{n-2}}. \tag{6′} \]

Thus, in all \(2j\) potentials \(u_{jh}(P)\) belong to the \(j\)-th class \((h=1,2,\ldots,2j)\). The first index denotes the number of the class, the second the order of the potential in the \(j\)-th class. For example, the potentials of the simple and double layers belong to the first class.

Let us consider the limiting properties of the potentials used, when the point \(P\) approaches the boundary point \(P_0\).

For simplicity of presentation we shall assume \(n=3\). We agree to denote:

\[ u_{jh,i}^{(2k)}(P_0)= \lim_{\substack{P\to P_0\\ P\in D}} \frac{\partial}{\partial \nu}\Delta^{k-1}u_{jh}(P); \]

\[ u_{jh,e}^{(2k)}(P_0)=\lim_{\substack{P\to P_0\\ P\in D_e}}\frac{\partial}{\partial \nu}\Delta^{k-1}u_{jh}(P); \]

\[ u_{jh,i}^{(2k+1)}(P_0)=\lim_{\substack{P\to P_0\\ P\in D}}\Delta^k u_{jh}(P); \]

\[ u_{jh,e}^{(2k+1)}(P_0)=\lim_{\substack{P\to P_0\\ P\in D_e}}\Delta^k u_{jh}(P). \]

Theorem 1. If the density of the potential \(u_{jh}(P)\), the function \(\mu_{jh}(Q)\), belongs to the class \(H_{2l}\), and \(\Sigma\in \Lambda_{2l+1}\), then

\[ u_{jh,i}^{(h+2l)}(P_0)-u_{jh,e}^{(h+2l)}(P_0)=L_{jh}^{(h+2l)}(\mu_{jh}); \tag{7} \]

if the density of the potential \(\mu_{jh}(Q)\in H_{2l+1}\), and \(\Sigma\in \Lambda_{2l+2}\), then

\[ u_{jh,i}^{(h+2l+1)}(P_0)-u_{jh,e}^{(h+2l+1)}(P_0)=L_{jh}^{(h+2l+1)}(\mu_{jh}), \tag{8} \]

where \(L_{jh}^{(h+2l)}\) and \(L_{jh}^{(h+2l+1)}\) are certain linear differential operators of order not exceeding \(2l\) in the densities \(\mu_{jh}(Q)\), in which the derivatives of \(\mu_{jh}(Q)\) are taken along the principal directions on the surface.

The proof is analogous to that given for the biharmonic equation in [4].

Remark 1. Under the assumptions made, the limiting equalities (7) and (8) are written down analogously for a space of arbitrary dimension; moreover, the order of the derivatives does not depend on the dimension of the space.

Let us consider the direct values of the potentials \(u_{jh}(P)\), i.e., the values at points of the surface \(P_0\); we shall denote them by \(\widetilde u_{jh}(P_0)\); the kernels of the direct values, respectively, by \(\widetilde\varphi_{jh}(P_0,Q)\).

It is obvious that, if \(l\le j\), then \(\widetilde u_{jh}^{(l)}(P_0)\) converges absolutely on the surface \(\Sigma\). When \(l>j\) and \(h\le j\), the kernels of the direct values as \(R(P_0,Q)\to 0\) are quantities of the following orders:

\[ \widetilde\varphi_{jh}^{(j+1)}(P_0,Q)=O\!\left(\frac{1}{R^3}\right),\qquad \widetilde\varphi_{jh}^{(j+2)}(P_0,Q)=O\!\left(\frac{1}{R^5}\right),\ldots \tag{9} \]

\[ \ldots,\quad \widetilde\varphi_{jh}^{(2j-h+1)}(P_0,Q)=O\!\left(\frac{1}{R^{2j-2h+3}}\right),\qquad \widetilde\varphi^{(2j-h+2)}(P_0,Q)= \]

\[ =O\!\left(\frac{1}{R^{2j-2h+3}}\right),\ldots,\qquad \widetilde\varphi_{jh}^{(2j)}(P_0,Q)=O\!\left(\frac{1}{R^{\,2j-2[h/2]+1}}\right). \]

Let us consider the direct values of the potentials \(\widetilde u_{jh}^{(l)}(P_0)\) for \(h>j\). For \(l<h\), it follows from (5) that \(u_{jh}^{(l)}(P)\) behaves in the same way as a potential with kernel

\[ \frac{\partial^{\,2\left[\frac{h-1}{2}\right]-h}} {\partial \nu^{\,2\left[\frac{h-1}{2}\right]-h}} \,R^{\,2\left[\frac{h+1}{2}\right]-3}. \]

The same also applies to \(\widetilde u_{jh}^{(l)}(P_0)\).

From (6) and (6′) it is seen that for

\[ h\le l\le h+j-\left[\frac{h+1}{2}\right] \]

\(\widetilde u_{jh}^{(l)}(P_0)\) are regular on \(\Sigma\).

For \(l>h+j-\left[\dfrac{h+1}{2}\right]\), the kernels of the direct values as \(R(P_0,Q)\to 0\) are quantities of order:

\[ \widetilde{\varphi}_{jh}^{(h+j-k)}(P_0,Q) =O\!\left(\frac{1}{R^3}\right),\ldots,\quad \widetilde{\varphi}_{jh}^{(2j-1)}(P_0,Q) =O\!\left(\frac{1}{R^{2j-2k-1}}\right), \]

\[ \widetilde{\varphi}_{jh}^{(2j)}(P_0,Q) =O\!\left(\frac{1}{R^{2j-2k+1}}\right) \quad (h=2k+1); \]

\[ \widetilde{\varphi}_{jh}^{(h+j-k)}(P_0,Q) =O\!\left(\frac{1}{R^3}\right),\ldots,\quad \widetilde{\varphi}_{jh}^{(2j-1)}(P_0,Q) =O\!\left(\frac{1}{R^{2j-2k-1}}\right), \]

\[ \widetilde{\varphi}_{jh}^{(2j)}(P_0,Q) =O\!\left(\frac{1}{R^{2j-2k-1}}\right) \quad (h=2k+2), \tag{10} \]

III. From (4) and (10) it follows that not for all \(l\le 2j\) do the
\(\widetilde{u}_{jh}^{(l)}(P_0)\) converge on \(\Sigma\); therefore they must be regularized.

Theorem 2. Let the density \(\mu_{jh}(Q)\) of the potential

\[ u_{jh}^{(l)}(P)=\int_{\Sigma}\mu_{jh}(Q)K(P,Q)\,d\sigma_Q \]

belong to the class \(H_{2h_1}\), and let the surface \(\Sigma\in\Lambda_{2h_1+2}\). Then, if \(K(P,Q)\) has a singularity of order

\[ \frac{1}{R^{2h_1}(P,Q)} \]

as \(P\to P_0\), the potential \(u_{jh}^{(l)}(P)\) can be represented as a sum of potentials with regular kernels and densities respectively

\[ \mu_{jh}(Q),\quad \frac{\partial \mu_{jh}(Q)}{\partial s_1},\quad \frac{\partial \mu_{jh}(Q)}{\partial s_2},\ldots,\quad \frac{\partial^{2h_1}\mu_{jh}(Q)}{\partial s_2^{2h_1}}. \]

The validity of the theorem follows from the following lemmas.

Lemma 1. If \(\Sigma\in\Lambda_{2h+2}\), then kernels of the form

\[ \frac{\cos^k\Theta(P,Q)}{R^{k+2h+1}(P,Q)} \]

can be represented as a linear combination of the functions

\[ \frac{\cos^j\Theta(P,Q)}{R^{j+1}(P,Q)} \quad (j=0,1,\ldots,2h+k) \]

and of derivatives of the latter up to order \(2h\) inclusive along the principal directions of the surface \(\Sigma\) at the point \(Q\).

Lemma 2. If \(\Sigma\in\Lambda_{2h+2}\), then kernels of the form

\[ \frac{\cos\Theta_0(P_0,Q)}{R^{2k+2}(P_0,Q)} \quad (P_0\in\Sigma,\; Q\in\Sigma,\; \cos\Theta_0=\cos(\widehat{\vec R,\vec \nu_P})) \]

can be represented as a linear combination of the functions

\[ \frac{\partial^{l_1+l_2}}{\partial s_1^{l_1}\partial s_2^{l_2}}\, \frac{\partial}{\partial \nu_P}\, \frac{\cos^j\Theta(P_0,Q)}{R^{j-1}(P_0,Q)} \quad (j=0,1,\ldots,2h+k;\; l_1+l_2=1,2,\ldots,2k). \]

Lemma 3. If \(\Sigma\in\Lambda_{2r+2}\), then the derivative of order \(2r\) along the principal directions of the kernel

\[ \frac{\cos^k\Theta(P_0,Q)}{R^{k+1}(P_0,Q)} \]

at the point \(P_0\) \((P_0\in\Sigma,\; Q\in\Sigma)\) can be represented as a linear combination of derivatives, along the principal directions at the point \(Q\), of regular kernels up to order \(2r\) inclusive.

Lemmas 1–3 are verified directly by means of Frenet’s formulas, by induction.

Lemma 4. If \(\Sigma \in \Lambda_1\) and \(\mu(Q)\in H_1\), then the potential

\[ \frac{1}{\pi}\int\limits_{\Sigma}\mu(Q)\frac{\partial}{\partial s_i}K_1(P,Q)\,d\sigma_Q \]

\((i=1,2)\) can be represented in the form

\[ -\frac{1}{\pi}\int\limits_{\Sigma}\mu(Q)\frac{\partial}{\partial s_i}K_1(P,Q)\,d\sigma_Q = \frac{1}{\pi}\int\limits_{\Sigma} \left\{ \mu(Q)a_0(Q)+ \frac{\partial\mu}{\partial s_1}(Q)a_1(Q)+ \right. \]

\[ \left. + \frac{\partial\mu}{\partial s_2}(Q)a_2(Q) \right\} K_1(P,Q)\,d\sigma_Q . \]

IV. Let us consider the boundary value problem \(m_{r_1,r_2,\ldots,r_n}\) with boundary conditions

\[ \sum_{j=1}^{2\left[\frac{k-1}{2}\right]} a_{kj}(P_0)u^{(j)}(P_0) + \sum_{j=2\left[\frac{k-1}{2}\right]+1}^{2k} b_{kj}(P_0)u^{(j)}(P_0) = f_k(P_0) \]

\[ (k=1,2,\ldots,m), \tag{11} \]

where the coefficients \(a_{kj}(P_0)\in H_{2m}\); concerning the coefficients \(b_{kj}(P_0)\in H_{2m}\) it is assumed that in each boundary condition at least one of them is different from 0.

Depending on which of the coefficients are different from 0, we obtain problems of different types. For definiteness we shall assume that \(r_1=r_2=1\), \(r_3=2\), \(r_4=3,\ldots\), \(r_{k_1}=k_1-1\), \(r_{k_1+1}=r_{k_1+2}=k_1\), \(r_{k_1+3}=k_1+1,\ldots\), \(r_{k_2}=k_2-2\), \(r_{k_2+1}=k_2-1\), \(r_{k_2+2}=k_2+1,\ldots\), \(r_{k_3+1}=r_{k_3+2}=k_3\), \(r_{k_3+3}=k_3+1,\ldots\), \(r_{k_4+1}=k_4-1\), \(r_{k_4+2}=k_4+2,\ldots\), \(r_m=m\) (\(r_j\) is the order of the \(j\)-th boundary condition).

In the problem under consideration there occur three pairs of boundary conditions of the same order: \(r_1=r_2=1\), \(r_{k_1+1}=r_{k_1+2}=k_1\), \(r_{k_3+1}=r_{k_3+2}=k_3\), and, respectively, boundary conditions of also three orders are absent: \(k_2\), \(k_4\), \(k_1+1\). Moreover, there may be any number of such pairs, less than or equal to \(\left[\frac{m}{2}\right]\) (for example, the Dirichlet problem or the principal boundary value problem is obtained when the number of pairs is equal to \(\left[\frac{m}{2}\right]\)).

We establish the following rule for choosing potentials. As the first \(k_1\) potentials we choose potentials of the \(k_2\)-class (the number of the class coincides with the number of the first omitted order) in such a way that each of them has a principal jump, in the term of highest order, in one of the limiting equalities. Next, the following \(k_3-k_1\) potentials are taken from class \(k_4\) (again the potentials are chosen so that the principal jump in the limiting equality is admitted by the term of highest order); the further \(k_4+1-k_3\) potentials belong to the class \(k_4+1\). The remaining potentials are potentials with kernels of the type \(R^h\) and \(\dfrac{\partial}{\partial\nu}R^h\), which ensure jumps in the remaining limiting equalities.

Remark 2. If in the boundary value problem there are several pairs of boundary conditions of one order, which are arranged one after another, for example

\[ r_1=r_2=1;\quad r_3=r_4=2,\ldots,\quad r_{2l_1-1}=r_{2l_1}=l_1,\quad r_{2l_1+1}=l_1+1,\ldots,\quad r_{2l_1-1+l_2}=r_{2l_1+l_2}=l_1+l_2, \]

then the corresponding \(2l_1+l_2\)

potentials can be chosen from the class whose number coincides with the number of the \(l_1\)-th omitted order. Hence it follows, for example, that the potentials of the Dirichlet problem are taken from the \(m\)-th class.

Remark 3. The recommended method of choosing potentials is not the only possible one; there is a certain arbitrariness in the choice of potentials. We transform each of the chosen potentials in order to simplify the limiting equalities.

If in the boundary-value problem there occur two boundary conditions of the same order (in our problem there are three such pairs), then one of the corresponding potentials will admit a jump in two limiting equalities, namely in the principal and limiting equalities of the omitted order. We again transform a potential of the \(j\)-th class by means of potentials of the \(j\)-th class.

Denote the new potentials by \(v_{jh}(\mu_{jh}; P)\):

\[ v_{jh}(\mu_{jh}; P)=\sum_{k=0}^{2j-h} u_{j,h+k}\left(\mu_{jh}^{\left(2\left[\frac{k}{2}\right]\right)}; P\right). \tag{12} \]

The densities of the potentials \(u_{j,h+k}\) are linear combinations of \(\mu_{jh}(Q)\) and derivatives of \(\mu_{jh}(Q)\) up to order \(2\left[\dfrac{k}{2}\right]\). For example, for the potential \(v_{k_2,2}(P)\) the following limiting equalities hold:

\[ v_{k_2,2,i}^{(2)}(P_0)-v_{k_2,2,e}^{(2)}(P_0)=c_2\mu_{k_2,2}(P_0), \]

\[ v_{k_2,2,i}^{(2k_2)}(P_0)-v_{k_2,2,e}^{(2k_2)}(P_0)=M_{k_2,2}(\mu_{k_2,2}), \tag{13} \]

\[ v_{k_2,2,i}^{(2l-1)}(P_0)-v_{k_2,2,e}^{(2l-1)}(P_0)=0, \]

where \(M_{k_2,2}\) is a linear combination with constant coefficients of the densities \(\mu_{k_2,2}(Q)\) and expressions of the form \(\Delta \mu_{k_2,2}(Q), \ldots, \Delta^{k_2-1}\mu_{k_2,2}(Q)\):

\[ M_{k_2,2}(\mu_{k_2,2})= a_0\Delta^{k_2-1}\mu_{k_2,2} +a_1\Delta^{k_2-2}\mu_{k_2,2} +\cdots+a_{k_2-1}\mu_{k_2,2}. \]

This combination is composed in such a way that, from the identical equality to zero of the operator \(M_{k_2,2}\) on the surface \(\Sigma\), there would follow the equality to zero of the function \(\mu_{k_2,2}(Q)\) \((\Sigma\subset \Lambda_{2m+2})\).

Let, for example, \(M\) be a second-order differential operator and

\[ M(\mu)=\Delta\mu+a\mu=0 \quad \text{on } \Sigma . \tag{14} \]

We show that the constant \(a\) can be chosen in such a way that from (14) there follows the equality to zero of the function \(\mu(Q)\) on \(\Sigma\):

\[ \int_{\Sigma}(\mu\Delta\mu+a\mu^2)\,d\sigma_Q=0. \]

\[ \Delta\mu=\frac{\partial^2\mu}{\partial s_1^2}+\frac{\partial^2\mu}{\partial s_2^2} =(D_x^2+D_y^2+D_z^2)\mu . \]

Using the integration-by-parts formula (see [1]), we obtain

\[ -\int_{\Sigma}\left[(D_x\mu)^2+(D_y\mu)^2+(D_z\mu)^2\right]\,d\sigma_Q+ \]

\[ +\int_{\Sigma}\left[6\varkappa^2-(D_x+D_y+D_z)\varkappa+a\right]\mu^2\,d\sigma_Q=0 \tag{15} \]

(\(\varkappa\) is the mean curvature of the surface \(\Sigma\) at the point \(Q\)).

Let \(a<-\max [6\chi^{2}-(D_x+D_y+D_z)\chi]\). Obviously, equality (15) is possible only in the case when \(\mu(Q)\equiv 0\).

The proof for the operator \(M_{k_2,2}\) is obtained by induction.

We shall seek the solution of the problem in the form (changing the notation for the densities)

\[ \begin{aligned} u(P)=&\, v_{k_2,1}(\mu_1;P)+v_{k_2,2}(\mu_2;P) +\sum_{l=3}^{k_1}\bigl[\delta_{1l}v_{k_2,2l-3}(\mu_l;P) +\delta_{2l}v_{k_2,2l-2}(\mu_l;P)\bigr]+ \\ &+v_{k_4,2k_1-1}(\mu_{k_1+1};P) +v_{k_4,2k_1}(\mu_{k_1+2};P)+ \tag{16}\\ &+\sum_{l=3}^{k_2-k_4+1} \bigl[\delta_{1l}v_{k_4,2k_1+2l-5}(\mu_{k_1+l};P) +\delta_{2l}v_{k_4,2k_1+2l-4}(\mu_{k_1+l};P)\bigr]+\\ &+\sum_{l=k_2-k_1+2}^{k_3-k_1} \bigl[\delta_{1l}v_{k_4,2k_1+2l-3}(\mu_{k_1+l};P) +\delta_{2l}v_{k_4,2k_1+2l-2}(\mu_{k_1+l};P)\bigr]+\\ &+v_{k_4+1,2k_3-1}(\mu_{k_3+1};P) +v_{k_4+1,2k_3}(\mu_{k_3+2};P)+\\ &+\sum_{l=3}^{k_4-k_3+1} \bigl[\delta_{1l}v_{k_4+1,2k_3+2l-5}(\mu_{k_3+l};P) +\delta_{2l}v_{k_4+1,2k_3+2l-4}(\mu_{k_3+2};P)\bigr]+\\ &+\sum_{l=k_4+2}^{m}\frac{1}{\pi}\int_{\Sigma} \mu_l(Q)\left[\delta_{1l}\frac{\partial}{\partial \nu_Q}R^{2l-3} +\delta_{2l}R^{2l-3}\right]\,d\sigma_Q, \end{aligned} \]

where \(v\) are transformed potentials; for any \(l\) either \(\delta_{1l}=1\) and \(\delta_{2l}=0\), or conversely. Obviously, the problem \(m_{r_1,r_2,\ldots,r_m}\) is reduced to the following system of integro-differential equations:

\[ \begin{aligned} &\sum_{j=1}^{2\left[\frac{k-1}{2}\right]} a_{kj}(P_0)\sum_{h=1}^{m}\widetilde v_{f,h}^{(j)}(P_0) +\sum_{j=2\left[\frac{k-1}{2}\right]+1}^{2k} b_{kj}(P_0)\sum_{h=1}^{m}\widetilde v_{f,h}^{(j)}(P_0)+\\ &\qquad\qquad +\alpha_k(P_0)\mu_k(P_0)=f_k(P_0) \quad (k=1,2,\ldots,m;\ f=k_2,k_4,k_4+1,m) \end{aligned} \tag{17} \]

(the derivatives of the unknown densities are contained in \(\widetilde v_{f,h}^{(j)}(P_0)\)).

System (17) is equivalent to a certain system of Fredholm integral equations of the second kind. Indeed, taking into account Theorem 1 and Lemmas 1–4, it is easy to show that the \(l\)-th equation of system (17) can be differentiated along the principal directions of the surface \(\Sigma\) \(l'\) times, where \(l'\) is the order of the highest derivative \(\mu_l(Q)\).

Let \(f_l(P_0)\in H^{l'}\). Then, if it is proved that the corresponding system of Fredholm integral equations of the second kind has a unique solution for any \(f_l(P_0)\in H^{l'}\) \((l=1,2,\ldots,m)\) in the class of bounded functions, the system of integro-differential equations (17) will have a unique solution \(\mu_l(Q)\) \((l=1,2,\ldots,m)\) in the class \(H^{l'}\). Thus it will be proved that the solution of the original problem \(m_{r_1,r_2,\ldots,r_m}\) exists for any functions \(f_l(P_0)\in H^{l'}\), belongs to the class \(H_{m+j}\) in \(D+\Sigma\), and can be represented in the form (16).

Remark 4. If \(\mu_l(Q)\) is the density of the potential \(v_{jh}(P)\) (\(j\) is the class number, \(h\) the number of the potential in the \(j\)-th class), then

\[ l' \le 2j-2\left(\left[\frac{h}{2}\right]\right). \]

V. We shall prove that system (17) is solvable, if the problem cannot have two solutions.

Consider the corresponding system of homogeneous equations. This system has only the trivial solution.

Let \(\mu_{10}(P_0), \mu_{20}(P_0), \ldots, \mu_{m0}(P_0)\) be a nonzero solution of the system of homogeneous equations. Consider the function \(u_0(P)\) which is obtained if in (16), instead of the densities \(\mu_j(Q)\), one substitutes \(\mu_{j0}(Q)\). It is obvious that \(u_0(P)\) is a solution of problem (1—2), when

\[ f_h(P_0)=0. \]

By uniqueness,

\[ u_0(P)\equiv 0 \quad \text{in } D. \]

Consequently,

\[ u_{0i}^{(l)}(P_0)=0;\quad l=1,\,2,\ldots,\,2m. \]

Consider this function in the exterior domain \(D_e\). In \(D_e\) it satisfies the equation \(\Delta^m u=0\) and the following boundary conditions:

\[ \begin{aligned} u_{0e}^{(1)}(P_0)&=-c_1\mu_{10}(P_0),\\ u_{0e}^{(2)}(P_0)&=-c_2\mu_{20}(P_0),\\ u_{0e}^{(3)}(P_0)&=-c_{31}\delta_{13}\mu_{30}(P_0),\\ u_{0e}^{(4)}(P_0)&=-c_{32}\delta_{23}\mu_{30}(P_0),\\ &\cdots\\ u_{0e}^{(2m-2k+1)}(P_0)&=\delta_{1,\,2m-2k+1}\sum_{l\le j<m} M_j\bigl(\Delta^{m-k-j+l}\mu_{j0},\,\Delta^{m-k-j+l-1}\mu_{j0},\,\ldots,\,\mu_{j0}\bigr),\\ u_{0e}^{(2m-2k+2)}(P_0)&=\delta_{2,\,2m-2k+1}\sum_{l\le j<m} N_j\bigl(\Delta^{m-k-j+l}\mu_{j0},\,\Delta^{m-k-j+l-1}\mu_{j0},\,\ldots,\,\mu_{j0}\bigr),\\ &\cdots\\ u_{0e}^{(2m-1)}(P_0)&=-c_{m1}\delta_{1m}\mu_{m0}(P_0),\\ u_{0e}^{(2m)}(P_0)&=-c_{m2}\delta_{2m}\mu_{m0}(P_0); \end{aligned} \tag{18} \]

\(\delta_{1j}\) is equal to 0 or 1, while \(\delta_{2j}\) is respectively equal to 1 or 0.

Introduce the function \(u_{01}(P)=\Delta^{m-1}u_0(P)\). This function satisfies the following conditions:

\[ \Delta u_{01}=0 \quad \text{in } D_e, \]

\[ u_{01}\big|_{\Sigma}=-c_{m1}\delta_{1m}\mu_{m0}(P_0), \]

\[ \frac{\partial u_{01}}{\partial \nu}\bigg|_{\Sigma}=-c_{m2}\delta_{2m}\mu_{m0}(P_0). \]

Thus, it is a solution either of the exterior Neumann problem with zero boundary condition \((\delta_{2m}=0)\), or, respectively, of the exterior Dirichlet problem \((\delta_{1m}=0)\). It is easy to see that \(u_{01}(P)=O\!\left(\frac{1}{r}\right)\) at infinity. Consequently, \(u_{01}(P)\equiv 0\) in \(D_e\). Hence, \(\mu_{m0}(Q)\equiv 0\) on \(\Sigma\). Next, the function \(u_{02}(P)=\Delta^{m-2}u_0(P)\) is considered. It also satisfies the exterior problem either of Neumann or Dirichlet type with zero boundary condition, and at infinity has order \(1/r\). Consequently, \(\Delta^{m-2}u_0(P)\equiv 0\) in \(D_e\) and \(\mu_{m-1,0}(Q)\equiv 0\) on \(\Sigma\).

Suppose that we have carried out \(k-1\) similar steps, i.e., have shown that \(\Delta^{m-(k-1)}u_0(P)\equiv 0\) in \(D_e\) and that the corresponding density is equal to 0 on the surface.

Consider the function \(u_{0k}(P)=\Delta^{m-k}u_0(P)\). Obviously, this function is harmonic and satisfies the zero boundary condition (either it itself vanishes on the surface, or its normal derivative does).

It is easy to show that \(u_{0k}(P)=O\left(\dfrac{1}{r}\right)\) at infinity.

Thus, \(u_{0k}(P)\) is a solution of the problem

\[ \Delta u_{0k}=0 \quad \text{in } D_e, \]

\[ u_{0k}\big|_{\Sigma}=0, \]

\[ u_{0k}=O\left(\frac{1}{r}\right) \quad \text{at } \infty . \]

(For definiteness, the Dirichlet problem is considered.) Moreover,

\[ \frac{\partial u_{0k}}{\partial \nu}(P_0) = N_l(\Delta^{m-k}\mu_{10},\, \Delta^{m-k-1}\mu_{10},\, \ldots,\, \mu_{l0}) \quad \text{on } \Sigma, \]

whence it follows that \(u_{0k}(P)\equiv 0\) in \(D_e\), and consequently,

\[ N_l(\Delta^{m-k}\mu_{10},\, \Delta^{m-k-1}\mu_{10},\, \ldots,\, \mu_{l0})\equiv 0 \quad \text{on } \Sigma. \]

But the combination \(N_l\) is composed in such a way that from its identical vanishing on the surface \(\Sigma\) it follows that \(\mu_{l0}(Q)\) is identically zero.

Thus, the homogeneous system has only the trivial solution, and, consequently, the system of nonhomogeneous integro-differential equations has a solution and it is unique; this means that the original boundary value problem has a solution.

Remark 5. The above proof of solvability of the boundary value problem for the equation \(\Delta^m u=0\) can be used to investigate similar problems also for the general equation

\[ \Delta^m u+a_1\Delta^{m-1}u+\ldots+a_m u=0. \]

References

1. Günter N. M. Potential theory and its application to the fundamental problems of mathematical physics. GTTI, 1953.

2. Vekua I. N. On metaharmonic functions. Proceedings of the Tbilisi Mathematical Institute, 12, 105–116, 1943.

3. Lopatinskii Ya. B. UMZh, 5, No. 2, 123–151, 1953.

4. Panich O. I. Izv. vuzov, Mathematics, No. 3 (22), 80–90, 1961; No. 4 (23), 1961; No. 6 (25), 1961; No. 1 (26), 1962.

5. Lobodzinska I. G. Proceedings of ODU named after Mechnikov, XCIV, 148, issue III, Collection of Young University Scholars, 1958.

6. Lobodzinska I. G. Scientific Yearbook of OSU named after Mechnikov, issue 2, 1951.

7. Lobodzinska I. G. Scientific Yearbook of OSU named after Mechnikov, issue 2, 1961.

8. Lobodzinska I. G. Proceedings of ODU named after Mechnikov, XCVII, 151, issue VI, 1961.

Received by the editors
December 1, 1965

Odessa State University
named after I. I. Mechnikov